nLab
perfect Morse function

One of the basic results of Morse theory are the so-called Morse inequalities involving Morse numbers. The values in the Morse inequalities depend on a choice of Morse function. The best possible case if when the Morse inequalities are equalities. In that case we speak of a perfect Morse function.

It has long been known that the existence of a perfect Morse function on every smooth 3-dimensional manifold is equivalent to the 3-dimensional Poincaré conjecture, which is now the Poincaré–Perelman theorem.